Non-degenerate mixed functions

Abstract

Mixed functions are analytic functions in variables z1, . . . , zn and their conjugatesz1, . . . ,zn. We introduce the notion of Newton nondegeneracy for mixed functions and develop a basic tool for the study of mixed hypersurface singularities. We show the existence of a canonical resolution of the singularity, and the existence of the Milnor fibration under the strong non-degeneracy condition.

Singular open book structures from real mappings

“…Let us at least point out two situations where one proves directly the weaker condition (3). One of them is contained in the proof of [17,Lemma 51] in the setting of mixed functions , where Oka shows the existence of a full tube fibration for a special class of mappings, namely the super strongly non-degenerate mixed functions. This is a condition which allows Oka to prove in [ /2 + h o t , which yields a sign contradiction.…”

“…It turns out that condition In order to produce a new class of higher dimensional purely real examples, we use the theory of mixed functions, the real analytic mappings R 2 C → C R 2 , recently developed by Mutsuo Oka (see [16,17] and our footnote at Section 4). The necessary definitions are given in Section 4.…”

Real map germs and higher open book structures

“…In a more recent preprint [12], Oka obtaines such a result in the particular class of "polar weighted homogeneous mixed functions". Our paper is an improved version of the manuscript [5] which has been made available to a group of mathematicians since early December 2007.…”

New open-book decompositions in singularity theory

“…These polynomials were introduced by Cisneros-Molina in [4] following ideas from Ruas, Seade and Verjovsky in [22] and studied by Oka in [16] and [17]. Let ( p 1 , .…”